RotatE embeds knowledge graphs using rotations in complex space for better link prediction.
problem Predicting missing links in knowledge graphs.
method RotatE models relations as rotations in complex vector space, using self-adversarial negative sampling.
result RotatE models and infers various relation patterns, outperforming existing models.
Wick rotations between pseudo-Riemannian spaces using real GIT.
problem Existence and non-existence of Wick-rotations between different pseudo-Riemannian spaces.
method Using real GIT to study structure groups of pseudo-Riemannian spaces as real forms of complex Lie groups.
result Derivation of new results regarding the existence and non-existence of Wick-rotations.
Solves scalar curvature equations for rotation invariant Kähler metrics on complex space.
problem Finding rotation invariant Kähler metrics with constant scalar curvature on complex space.
method Reduces the scalar curvature equation to a system of ODEs and solves them.
result Obtains complete lists of rotation invariant metrics with zero or positive scalar curvature in lower dimensions.
Classifies Kähler-Einstein manifolds with specific properties.
problem Classifying Kähler-Einstein manifolds with rotation invariant metrics.
method Uses Kähler immersions into complex projective spaces with Fubini-Study metrics.
result Classifies manifolds with codimension ≤ 3 and rotation invariant metrics.
We put in a general framework the situations in which a Riemannian manifold admits a family of compatible complex structures, including hyperkahler metrics and the Spin-rotations of arxiv:1302.2846. We determine the (polystable) holomorphic bundles which are rotable, i.e., they remain holomorphic when we change a compl…
New quadratic forms expand and rotate linear endomorphisms in geometric theory.
problem Understanding the expansion and rotation properties of linear endomorphisms.
method Constructing new quadratic forms based on two-plane rotations.
result Established relations among eigenvalues, eigendirections, and matrix invariants.
Proves embedding condition for certain 2-complexes in 3-space.
problem Embedding simply connected 2-complexes in 3-space.
method Analyzes link graphs and their embeddings in the plane.
result Trivial first homology group implies embedding in 3-space.
Solves classical problem with Kähler-Einstein metrics in complex projective spaces.
problem Classical problem of non-isometric bidimensional Kähler-Einstein submanifolds.
method Listed complete non-isometric bidimensional rotation invariant Kähler-Einstein submanifolds.
result Solves the classical problem in the specified case.
Quaternion embeddings model entities and relations in knowledge graphs.
problem Modeling latent inter-dependencies and expressive rotations in knowledge graphs.
method Quaternion-valued embeddings and rotations in hypercomplex space.
result Quaternion embeddings achieve state-of-the-art performance on knowledge graph benchmarks.
Harmonic Networks make images equivariant to translation and rotation.
problem Making CNNs equivariant to rotation.
method Replacing CNN filters with circular harmonics to achieve patch-wise translation and 360-degree rotation equivariance.
result Deep feature maps encode complicated rotational invariants.
Study classifies Kähler-Einstein metrics with rotational symmetries.
problem Classify Kähler-Einstein metrics with rotational symmetries.
method Focus on integrable structures to classify metrics.
result Classify Kähler-Einstein metrics with rotational symmetries.
Solves complex clustering and rotation synchronization problem.
problem Challenges in classifying and synchronizing rotated objects into multiple categories.
method Semidefinite programming relaxations to solve the joint problem of community detection and synchronization.
result Exact recovery of community detection and synchronization when extending stochastic block model.
New minimal surfaces in 4D space discovered using complex rotations.
problem Discovering new minimal surfaces in 4D space.
method Complex parabolic rotations of holomorphic null curves in 4C space.
result Existence of minimal surfaces foliated by conic sections in 4D space.
A new rotation invariant method for 3D medical imaging classification.
problem Computational expense and lack of rotation invariance in 3D medical image processing.
method Proposes a rotation invariant convolution operator using hypersphere topology.
result Demonstrates improved classification accuracy and rotation invariance.
2D twistor manifolds explain Teukolsky equations for rotating black holes.
problem Understanding the geometry of Teukolsky equations for rotating black holes.
method Using 2D twistor manifolds to explain the Teukolsky equations.
result All geometric structures can be understood by considering a 2D twistor manifold.
RotDCF decomposes CNN filters for rotation-equivariant deep networks.
problem Handling global deformations in images for vision tasks.
method Decomposes convolutional filters over joint steerable bases for rotation-equivariance.
result Significantly reduces model size and computational complexity while preserving performance.
Using gauge theory for Spin(7)-manifolds of dimension 8, we develop a procedure, called Spin-rotation, which transforms a (stable) holomorphic structure on a vector bundle over a complex torus of dimension 4 into a new holomorphic structure over a different complex torus. We show non-trivial examples of this procedure …
RotRNN uses rotations to simplify long sequence modelling.
problem Complex initialisation and normalisation schemes in linear recurrent models.
method RotRNN employs rotation matrices to simplify and normalise linear recurrent models.
result RotRNN achieves competitive performance on long sequence modelling datasets.
New method untangles double-twist loops efficiently.
problem Tangling of double-twist loops in 3D space.
method Geometrically defined waving procedure to untangle loops.
result Minimum complexity of untanglings is reduced.
New methods improve multi-agent reinforcement learning by addressing rotational dynamics.
problem Reproducibility crisis in multi-agent reinforcement learning.
method Reframing MARL approaches using Variational Inequalities (VIs) and proposing gradient-based VI methods.
result Significant performance improvements across benchmarks, including better convergence to equilibrium strategies in zero-sum games.
The paper proves mirror symmetry for del Pezzo surfaces and computes related structures.
problem Understanding mirror symmetry for del Pezzo surfaces and related geometric structures.
method Using hyperKähler rotation and Floer theory, the paper constructs and compares Landau-Ginzburg mirrors and complex affine structures.
result The limit of the complex affine structure of special Lagrangian fibrations agrees with integral affine structures.
Capsule networks improve classification accuracy on complex data.
problem CNN's failure to account for spatial hierarchies and lack of rotational invariance.
method Dynamic routing and reconstruction regularization to create capsules that are both rotation invariant and spatially aware.
result Capsule networks achieve a state-of-the-art 0.25% test error on MNIST without data augmentation.
TURB-Rot provides a large database of turbulent rotating flow snapshots for research.
problem Lack of large-scale, high-resolution datasets for turbulent rotating flows.
method Direct Numerical Simulations of Navier-Stokes equations with rotation.
result Provides a diverse set of 300K complex images and fields for testing.
Study on rotating surfaces in 4D space with matrices.
problem Understanding rotational surfaces in pseudo-Euclidean 4-space.
method Defined hyperbolic and elliptic rotational surfaces using curves and matrices in 4D semi-Euclidean space.
result Generated rotated surfaces using specific rotation matrices.
Abstract: Generalizes supergravity c-map to quaternionic manifolds.
problem Construct quaternionic manifolds from hypercomplex manifolds.
method Construct conical hypercomplex manifolds and associate quaternionic manifolds.
result Quaternionic manifolds can be associated to special complex manifolds.
A new transform links rotating calorons to solutions of a differential equation.
problem Existence and characterization of rotating calorons.
method Formulated a Nahm transform to relate rotating calorons to solutions of a delayed-differential equation.
result Existence of an eight-parameter family of rotating calorons with nontrivial holonomy.
We define general rotational surfaces of elliptic and hyperbolic type in the pseudo-Euclidean 4-space with neutral metric which are analogous to the general rotational surfaces of C. Moore in the Euclidean 4-space. We study Lorentz general rotational surfaces with plane meridian curves and give the complete classificat…
The paper defines and analyzes homotopic rotation sets for surfaces of higher genus.
problem Defining and analyzing homotopic rotation sets for surfaces of higher genus.
method Developed a definition and proved several results using the theory of Le Calvez and Tal.
result Found that the homotopic rotation set can imply the existence of infinitely many periodic orbits under certain conditions.
The paper develops RM frames for isotropic and pseudo-isotropic spaces and characterizes spherical curves.
problem Differential geometry of curves in isotropic and pseudo-isotropic 3-spaces.
method Developing rotation minimizing frames and applying them to spherical curves in isotropic and pseudo-isotropic spaces.
result Characterization of spherical curves via linear equations involving curvatures and RM frames.
Study of timelike surfaces in Minkowski space with specific geometric properties.
problem Characterizing geometric properties of timelike surfaces in Minkowski space.
method Analytical study of two types of timelike general rotational surfaces.
result Explicit descriptions of minimal and surfaces with specific curvature properties.
The study characterizes loxodromes on specific rotational surfaces in 3D space.
problem Characterizing loxodromes on rotational surfaces with special geometric properties.
method Parametrizations and curvature/torsion calculations for loxodromes on various rotational surfaces.
result The loxodrome on a flat rotational surface is a general helix.
Overview of methods for rotating 2D and 3D data.
problem Processing data with equivariance/invariance under rotations.
method An overview of methods for 2D and 3D rotations.
result Identification of commonalities and links between methods.
Enhanced rotation prediction improves SSL models by capturing both shape and texture information.
problem Rotation prediction misses texture information, limiting model performance.
method Introduces image enhanced rotation prediction (IE-Rot) that combines rotation and image enhancement tasks.
result IE-Rot models outperform Rotation on various benchmarks.
General rotational surfaces as a source of examples of surfaces in the four-dimensional Euclidean space have been introduced by C. Moore. In this paper we consider the analogue of these surfaces in the Minkowski 4-space. On the base of our invariant theory of spacelike surfaces we study general rotational surfaces with…
We introduce Group equivariant Convolutional Neural Networks (G-CNNs), a natural generalization of convolutional neural networks that reduces sample complexity by exploiting symmetries. G-CNNs use G-convolutions, a new type of layer that enjoys a substantially higher degree of weight sharing than regular convolution la…
The paper introduces REQNNs for robust 3D point cloud processing.
problem 3D point cloud processing robustness to rotations.
method Revised neural networks using quaternion features for rotation-equivariance.
result REQNNs exhibit higher rotation robustness compared to original networks.
Rotation systems can't always be drawn in surfaces.
problem Rotation systems and simple drawings in surfaces.
method Extended the plane result to all fixed surfaces.
result Existence of rotation systems not arising from simple drawings in any fixed surface.
RotEqNet preserves rotation symmetry in fluid systems using high-order tensors.
problem Lack of rotational symmetry in machine learning models for fluid systems.
method Introduces RotEqNet, a network that guarantees rotation-equivariance for high-order tensors.
result RotEqNet reduces errors and maintains rotation-equivariance in fluid systems.
Paper learns to rotate filters for group convolutions.
problem Difficult to rotate 3x3 filters on pixel grids.
method Learn filter basis and rotation-invariant coefficients; switch basis for rotation.
result Produces feature maps insensitive to input rotations.
Unified DNN-based precoder for MIMO networks with multiple objectives.
problem Optimizing data transmission, energy harvesting, and security in MIMO networks.
method Rotation-based precoding and DNN for multi-objective optimization.
result DNN-based precoder reduces computational complexity and achieves near-optimal performance.
The paper applies Clairaut's theorem to rotational surfaces in pseudo-Euclidean 4-space.
problem Exploring geodesic curves on rotational surfaces in pseudo-Euclidean 4-space.
method Expressing Clairaut's theorem and deriving equations for geodesic curves.
result Characterization of geodesic curves on hyperbolic and elliptic surfaces of rotation.
Study on knots formed by Coxeter galleries, finding bounds and symmetric trefoils.
problem Understanding knots created by Coxeter galleries.
method Examined knots in affine Coxeter complex of type \widewedge{B3}, constructing galleries and proving properties.
result Found bounds on stick number and smallest length of symmetric trefoils.
We consider n-dimensional discrete motions such that any two neighbouring positions correspond in a pure rotation ("rotating motions"). In the Study quadric model of Euclidean displacements these motions correspond to quadrilateral nets with edges contained in the Study quadric ("rotation nets"). The main focus of ou…
Minimal sets of moves for rotational Reidemeister diagrams are identified.
problem Understanding the minimal sets of moves for rotational Reidemeister diagrams.
method Detailed description and proof of minimal generating sets for rotational Reidemeister moves.
result Minimal generating sets for oriented, framed links contain 5 moves.
Study Wick-rotations of Lie groups using GIT results.
problem Understanding Wick-rotations of pseudo-Riemannian Lie groups.
method Using results from real GIT, analyze invariant metrics and Lie algebras.
result Existence and conjugacy of Cartan involutions for pseudo-Riemannian Lie groups.
Study real GIT and Wick-rotations of pseudo-Riemannian manifolds.
problem Understanding Wick-rotations and their conditions for pseudo-Riemannian manifolds.
method Extending earlier results, providing sufficient and necessary conditions for Wick-rotatability, and deriving an invariance theorem.
result Derived sufficient and necessary conditions for pseudo-Riemannian manifolds to be Wick-rotatable.
The space Z of leftinvariant orthogonal almost complex structures, keeping the orientation, on 6-dimensional Lie groups is researched. To get explicit view of this space elements the isomorphism of Z and CP3 is used. The explicit formula for arbitrary leftinvariant orthogonal almost …
New method studies moving points on curves using rotating frames.
problem Understanding the motion of points on curves.
method Constructing rotating frames for curves and analyzing the motion of points within these frames.
result A new binary mathematical formation mechanism for curves based on linear and rotational motion.